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Difference between revisions of "2023 AMC 12B Problems/Problem 2"
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<math>\textbf{(A) }\$46\qquad\textbf{(B) }\$50\qquad\textbf{(C) }\$48\qquad\textbf{(D) }\$47\qquad\textbf{(E) }\$49 </math>
 
<math>\textbf{(A) }\$46\qquad\textbf{(B) }\$50\qquad\textbf{(C) }\$48\qquad\textbf{(D) }\$47\qquad\textbf{(E) }\$49 </math>
 +
 +
==Solution 1==
 +
We can create the equation:
 +
<cmath>0.8x \cdot 1.075 = 43</cmath>
 +
using the information given. This is because x, the original price, got reduced by 20%, or multiplied by 0.8, and it also got multiplied by 1.075 on the discounted price. Solving that equation, we get
 +
<cmath>\frac{4}{5} \cdot x \cdot \frac{43}{40} = 43</cmath>
 +
<cmath>\frac{4}{5} \cdot x \cdot \frac{1}{40} = 1</cmath>
 +
<cmath>\frac{1}{5} \cdot x \cdot \frac{1}{10} = 1</cmath>
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<cmath>x  = \boxed{50}</cmath>
 +
 +
~lprado

Revision as of 17:40, 15 November 2023

Problem

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $$43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?


$\textbf{(A) }$46\qquad\textbf{(B) }$50\qquad\textbf{(C) }$48\qquad\textbf{(D) }$47\qquad\textbf{(E) }$49$

Solution 1

We can create the equation: \[0.8x \cdot 1.075 = 43\] using the information given. This is because x, the original price, got reduced by 20%, or multiplied by 0.8, and it also got multiplied by 1.075 on the discounted price. Solving that equation, we get \[\frac{4}{5} \cdot x \cdot \frac{43}{40} = 43\] \[\frac{4}{5} \cdot x \cdot \frac{1}{40} = 1\] \[\frac{1}{5} \cdot x \cdot \frac{1}{10} = 1\] \[x = \boxed{50}\]

~lprado

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